On Wronskians and $qq$-systems

TL;DR

This paper introduces a geometric perspective on $qq$-systems related to the twisted Gaudin model, using $G$-Wronskians and their connection to twisted $G$-oper connections, revealing new structural insights.

Contribution

It presents a novel geometric framework for understanding $qq$-systems via $G$-Wronskians and their relation to twisted $G$-oper connections, advancing the theoretical foundation.

Findings

$qq$-systems are realized as relations between generalized minors of $G$-Wronskians.

Established a link between $G$-Wronskians and twisted $G$-oper connections.

Provided a new geometric interpretation of the Bethe ansatz equations for the twisted Gaudin model.

Abstract

We discuss the $qq$-systems, the functional form of the Bethe ansatz equations for the twisted Gaudin model from a new geometric point of view. We use a concept of $G$-Wronskians, which are certain meromorphic sections of principal $G$-bundles on the projective line. In this context, the $qq$-system, similar to its difference analog, is realized as the relation between generalized minors of the $G$-Wronskian. We explain the link between $G$-Wronskians and twisted $G$-oper connections, which are the traditional source for the $qq$-systems.